{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Regularised regression</h3>\n",
    "<p>Our original squared loss function in matrix/vector notation is:\n",
    "$$ L = \\frac{1}{N} (\\mathbf{t} - \\mathbf{X}\\mathbf{w})^T(\\mathbf{t} - \\mathbf{X}\\mathbf{w}) $$\n",
    "Here's another loss function:\n",
    "$$ L = \\lambda \\mathbf{w}^T\\mathbf{w} + \\frac{1}{N}(\\mathbf{t} - \\mathbf{X}\\mathbf{w})^T(\\mathbf{t} - \\mathbf{X}\\mathbf{w}) $$\n",
    "Recall that we're minimising this function and so (if $\\lambda>0$) this additional term will penalise large positive and negative values in $\\mathbf{w}$. $\\lambda$ controls how much influence this new term has over the original squared error term.</p>\n",
    "\n",
    "<p>Differentiating this with respect to $\\mathbf{w}$ and then setting to zero (this is a good exercise to do) results in:\n",
    "$$ (\\mathbf{X}^T\\mathbf{X} + N\\lambda\\mathbf{I})\\mathbf{w} = \\mathbf{X}^T\\mathbf{t} $$\n",
    "where $\\mathbf{I}$ is a square matrix with ones on the diagonal and zeros elsewhere (the identity matrix).</p>\n",
    "\n",
    "<p>To demonstrate the effect of this additional term, we will generate some synthetic data by using a quadratic function and assing some random (normal / Gaussian) noise.</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10580d550>]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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kSeTl5QEwfPhwRowYQdOmTT1OJlJzqUCIVBIIBOiVmclwv5+xoRAGOGDe1Kn0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\nYJ7wjWYvAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10551f390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "x = np.linspace(0,10,10)[:,None]\n",
    "y = x**2 - 3*x + 4\n",
    "t = y + np.random.normal(0,10,(x.size,1))\n",
    "import pylab as plt\n",
    "%matplotlib inline\n",
    "plt.plot(x,y,'k')\n",
    "plt.plot(x,t,'ro')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h4>Creating $\\mathbf{X}$</h4>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "maxorder = 7\n",
    "x_test = np.linspace(0,10,30)[:,None]\n",
    "X = np.ones_like(x)\n",
    "X_test = np.ones_like(x_test)\n",
    "for i in range(1,maxorder+1):\n",
    "    X = np.hstack((X,x**i))\n",
    "    X_test = np.hstack((X_test,x_test**i))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h4>Loop over different values of $\\lambda$</h4>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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fTGZocTGhUOaEiA4d/ERT55wT23f55b5n/tq1wdUl65o/34+i2Xln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\nAVsClWZWa2a1wMHABWa2OtISJMn1FVB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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1058d3dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x105c46650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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k1dWF4HDJJfDKK+mvxOi233z+uuMYrvl/02lfW8vn5eX0HjSIaePHE4vFoiq5VVGAEBGR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\nOAJEAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105c88a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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XXAEvv9y4/Wc/8xMk99gjmLpyhUaCREQko3z6qV+3oaSkcXg49FB44w2YOlXhIR1SHiDM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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105d44d10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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45JBo6xKJigKEiMhG/OtfYY2Hd98Nx9ttF5am/va3o61LJEqahSEi0ownngiL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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105ddd990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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7D44/Hl57LRy3bw8PPAADB0Zbl0hUFCBERNbjlVdgwAD48MNwvMUWYY2H3r2j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\nPBqGAZpra+r75tpS69iQ7xs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      "text/plain": [
       "<matplotlib.figure.Figure at 0x105e650d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for lamb in [0,0.01,0.1,1,10,100]:\n",
    "    w = np.linalg.solve(np.dot(X.T,X) + x.size*lamb*np.identity(maxorder+1),np.dot(X.T,t))\n",
    "    f_test = np.dot(X_test,w)\n",
    "    plt.figure()\n",
    "    plt.plot(x_test,f_test,'b-',linewidth=2)\n",
    "    plt.plot(x,t,'ro')\n",
    "    title = '$\\lambda=$%g'%lamb\n",
    "    plt.title(title)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>As $\\lambda$ increases, high values in $\\mathbf{w}$ are more heavily penalised which leads to *simpler* functions. Why do lower values correspond to simpler functions?</p>\n",
    "<p>Firstly, what does *simpler* mean?</p>\n",
    "<p>I would argue that simpler functions have smaller derivatives (first, second, etc) as they typically change more slowly. In our polynomials, the derivatives are dependent on the values of $\\mathbf{w}$. In particular our polynomial is:\n",
    "$$ t = \\sum_{d=0}^D w_d x^d $$\n",
    "and the first derivative is:\n",
    "$$ \\frac{dt}{dx} = \\sum_{d=1}^D dw_d x^{d-1} $$\n",
    "and second is:\n",
    "$$ \\frac{d^2t}{dx^2} = \\sum_{d=2}^D d(d-1)w_d x^{d-2} $$\n",
    "which in both cases increases with increasing values of $w_d$. So penalising high (positive and negative) values decreases (in general) the gradients (and gradients of gradients, etc).</p> "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
